Skip to main content

Full text of "Men Of Mathematics"

See other formats


ARITHMETIC THE  SECOND
material career. He lived so long that although some of his
•work (his theory of irrational numbers, described presently)
had been familiar to all students of analysis for a generation
before his death, he himself had become almost a legend and
many classed him with the shadowy dead. Twelve years before
his death, Teubner's Calendar for Mathematicians listed Dede-
kind as having died on 4 September 1899, much to Dedekind's
amusement. The day9 4 September, might possibly prove to be
correct, he wrote to the editor, but the year certainly was
wrong. * According to my own memorandum I passed this day
in perfect health and enjoyed a very stimulating conversation
on '^system and theory"' with my luncheon guest and honoured
friend Georg Cantor of Halle.'
Dedekind's mathematical activity impinged almost wholly
on the domain of number in its widest sense. \Ve have space for
only two of his greatest achievements and we shall describe
first his fundamental contribution, that of the 'Dedekind cut%
to the theory of irrational numbers and hence to the founda-
tions of analysis. This being of the very first importance we may
recall briefly the nature of the matter. If a, & are common whole
numbers, the fraction a & is called a rational number; if no
whole numbers m, n exist such that a certain "number' A is
expressible as m/ra, then A* is called an irrational number. Thus
V2, Vs, A/6 are irrational numbers. If an irrational number be
expressed in the decimal notation the digits following the
decimal point exhibit no regularities - there is no *period* which
repeats, as in the decimal representations of a rational number,
say 13/11, = 1-18181S ... , where the '18' repeats indefinitely.
How then, if the representation is entirely lawless, are decimals
equivalent to irrationals to be defined, let alone manipulated?
Have we even any clear conception of what an irrational
number is? Eudosus thought he had, and Dedekind's definition
. of equality between numbers, rational or irrational, is identical
with that of Eudoxus (see Chapter 2).
If two rational numbers are equal, it is no doubt obvious that
their square roots are equal. Thus 2x3 and 6 are equal; so
also then are Vz x 3 and V&. But it is not obvious that Vi x
Vs = V2 X 3, and hence that V2 X Va *= Vi". The un-
573